How Do You Apply Impulse and Momentum Fluid Mechanics to Pipeline Leak Detection?

This guest blog post is part of a series written by Edward J. Farmer, PE, ISA Fellow and author of the ISA book Detecting Leaks in Pipelines. To download a free excerpt from Detecting Leaks in Pipelines, click here. If you would like more information on how to purchase the book, click this link. To read all posts on pipeline leak detection in this series, scroll to the bottom of this post for the link archive.

Detecting leaks using the Principle of Impulse and Momentum involves keeping track of momentum and the forces that produce acceleration. Recall that the short-form mathematical description for momentum in fluid mechanics is:

dv/dt = a = F/M

Essentially, the change in velocity, dv, over a period of time, dt, produces acceleration, a; which is equal to the applied force, F, divided by the mass, M, of the fluid involved. Note that this clearly shows that a lot of fluid, or a particularly dense (heavy) fluid will accelerate more slowly than a smaller volume of fluid or a less dense fluid of the same volume.

In fluid mechanics, force, F, usually comes from pressure (P) acting over the pipe area (A). The net force acting on the fluid is the difference between the pressures at each end, which we note as ΔP, Consequently,

F = ΔP · A

This, of course, takes us to the most useful form of this relationship in pipe flow mechanics which is:

dv/dt = ΔP·A / M

The mass, of course is determined by multiplying the fluid density times its volume. For a pipeline segment of area, A, and length L our equation becomes:

dv/dt = (P · A) / (ρ · A · L) which simplifies to:

dv/dt = P / (ρ · L)  wherein ρ is the density in the region of length L

Momentum is defined as mass times velocity, but in fluid transients, including impulses, it is usually more useful to work directly with velocity and mass than calculate the intermediate momentum value. Notice that the area, A, disappeared because the ratio of A to A is simply 1. That works as long as the area is everywhere the same within the calculation region. If a change in pipeline diameter, or a valve (other than an open full-port valve) is involved then it becomes necessary to include relevant area-of-flow in the calculations.

This presents a simple picture of what happens on a section of pipe within supply and control points. Conditions within this “control volume” may not be everywhere consistent so you must know what you are looking for and simplify cautiously with that in mind. Here, we are building toward explaining what happens to produce a surge or similar event on a pipeline within a defined control volume.

A simple problem illustrates why this is important. Think about a transmission pipeline with a pressure of P1 at the upstream end and P2 at the other. Assume the line is flowing with a homogeneous fluid at velocity v. Suddenly the downstream valve closes. Let’s think about this a bit.

A lot of fluid is moving along at the velocity of flow, v. The mass in the line upon reaching the suddenly closed valve has to transition from velocity v to velocity 0. All other things being equal, and considering that energy must be conserved, all that velocity at the upstream face of the now closed valve must make a transition to 0. What is the head associated with that? Whatever head existed at the downstream end when flowing increases from the conversion of velocity head into pressure head. Essentially H2 increases by:

ΔH2 = v² / (2g) where g is the acceleration due to gravity

Note that this head increase is calculated in length units, meters in the SI system or feet in the English system. It adds to the pressure head and elevation head already there. Depending on exactly what happens, pressure head could go up or down depending on the equipment and geometry, but the head derived from the velocity is usually the big item.

I recall working on a problem with some crude oil some 40 years ago in which the geometry produced a change of 50 psi per ft/sec change in velocity. At a flow rate of 10 feet per second the pressure at the valve face IN THAT SITUATION increased by 500 psi. Essentially the pressure at the valve face rose markedly creating a pressure transient that had to go somewhere. From work I did on my Physical Science master’s program, I knew something about wave mechanics and understood that a mechanical wave (e.g., a pressure wave) upon encountering a transient event, does three things:

• It tries to continue in the direction it was going, but that isn’t an option with a closed valve (unless the forces involved are sufficient to move the valve out of the way).
• It may reflect back toward the source of flow. Essentially, the pressure spike at the closed valve’s face produces a pressure pulse that travels back in the direction from whence it came.
• It scatters in a manner consistent with reflection mechanics, dispersing itself about the pipe.

Suddenly, the nice and uniform flow in our pipe is replaced with a series of conditions that include a lot of pressure waves traveling here and there, precipitated by the velocity change and directed by wave mechanics. This kind of thing is the source of what is referred to as “surge,” usually meaning large pressure changes resulting from process equipment causing changes in flow velocity.

What happens when this pressure wave makes it back to the upstream end of the pipeline? It depends, of course, on the conditions it encounters. Essentially, the pressure spike produces a force that acts on the fluid to produce a change in flow velocity. Suddenly, discerning what exactly is happening from a few pressure readings becomes very problematic. Exactly when the event was precipitated matters. So do the conditions the wave, in its various travels, experiences. It can be important to know the starting time of this event, all the distances involved, and the speed of sound in the fluid everywhere. It may be necessary to correlate pressure values with the precipitating event or with fresh new ones. There are some tricks used in signal processing that make it possible, but nothing that makes it easy. In trying to discern what is really happening on the pipeline one is reminded of the old television commercial for magnetic recording tape that posited the question, “Is It Live or Is It Memorex?”

To make the point quickly and obviously, this discussion is based on a massive event with few degrees of freedom including some we haven’t discussed (e.g., the pump may care about this pressure reflecting back up the pipeline). Also, a moving wave loses energy as it runs along the pipeline. The highest frequencies are affected the most, resulting in the nice, sharp edges of a pressure impulse degrading over distance into smooth, roundish rise and decay curves, all at lower amplitude.

The point is, not only is it hard to calculate conditions during such a transient, it is almost impossible to tell if things that can be observed result from the transient or other fully independent mechanical or hydraulic events, such as leaks, occurring in more-or-less the same time frame or equipment operations.

Victor L. Streeter and E. Benjamin Wylie wrote a book called Fluid Transients that explores a lot of factors in this area. I met Streeter while working on a pipeline with some complex transient conditions. In those days it was hard to get a computer to solve the set of equations that Streeter understood were descriptive of the problem. That can be done now but remains a serious piece of computer work. Simply put, it remains challenging, albeit now-possible, to model at least some fluid transient situations.

Why are we going through all this? This is important in understanding issues with monitoring gathering systems, or pipelines with several simultaneously serviced ends or entry points. Besides that, it’s one of the most interesting areas in fluid mechanics, insightfully explored by Streeter and Wylie.